MATH 277, AUTUMN 2016
HOMEWORK 4: SATURATION
DUE WEDNESDAY, NOVEMBER 16 AT 5PM
Throughout this problem set, let T denote a consistent theory, i.e. a consistent set
of sentences in some language L. When Γ(x) is a set of formulas in the free variable
x, say that the model M realizes Γ if there is an element a ∈ Dom(M ) such that
M |= ϕ(a) for all ϕ ∈ Γ. In this case, call a a realization of Γ in M .
(1) Consider M = (N; <) where the symbols have their usual interpretation.
Let D be a nonprinicipal ultrafilter on ω. Prove that M ω /D is uncountable.
Suggestion: Assume for a contradiction it is countable, and diagonalize.
(2) (a) Write down axioms for an algebraically closed field of characteristic p
(call this Tp ) and for an algebraically closed field of characteristic 0
(call this T0 ), in the language L = {+, ×, 0, 1}.
(b) For each prime p, choose Mp |= TpQ
. Let D be a nonprincipal ultrafilter
on the set P of primes. Let N = p∈P Mp /D. Prove that N |= T0 .
(3) We say that a theory is κ-categorical if whenever M |= T , N |= T and
|M | = |N | = κ, M ∼
= N.
(a) Suppose T has only infinite models. Suppose T is κ-categorical for
some infinite κ ≥ |L| + ℵ0 . Prove that T is complete.
(b) In item (a), why do we need to assume T has no finite models?
(4) Let Γ(x) be a set of formulas in the free variable x. Show that t.f.a.e.:
(a) T has a model which realizes Γ.
(b) Every finite subset of Γ is realized in some model of T .
(c) Some model of T realizes every finite subset of Γ.
(d) T ∪ {∃x(ϕ1 (x) ∧ · · · ∧ ϕn (x)) : n < ω, ϕ1 , . . . , ϕn ∈ Γ} is consistent.
(5) Let Σ(x) be a set of formulas in one free variable, closed under conjunction,
and realized in some model M |= T . Suppose that for some infinite κ and
all N ≡ M , Σ has fewer than κ realizations in N . Prove that for some
ϕ ∈ Σ and some finite n,



_
M |= ∃x1 . . . ∃xn ∀z ϕ(z) =⇒ 
z = xi  .
i≤n
Date: Assignments are due in the course mailbox in the basement of Eckhart by 5pm on the
due date. No late homework assignments will be accepted; we will automatically drop the lowest
homework score. Homework may also be turned in in class or at office hours prior to the due date.
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MATH 277, AUTUMN 2016
HOMEWORK 4: SATURATION
Challenge problem (optional; turn in on a separate page, with your name on it):
Analyze the problem of showing that a nonprincipal ultrapower (where D is an
ultrafilter on some uncountable set) of a countable model is uncountable. What
about the ultrafilter are you really using in problem 1? (This is an exploratory
problem; I’ll give credit for any reasonable and interesting approaches.)